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From three-photon GHZ states to ballistic universal quantum computation
Mercedes Gimeno-Segovia, Pete Shadbolt, Dan E. Browne, Terry Rudolph
TL;DR
The paper addresses the resource demands of building renormalized cluster states for quantum computing. It proposes a lattice-renormalization scheme using deterministic 3-photon GHZ states and Bell pairs, achieving linear resource scaling, at least an order-of-magnitude resource reduction, and improved loss tolerance relative to a previous scheme.
Problem
Machine size, measured by required components and resources, is a major concern in designing feasible quantum-computing architectures.
Method
The scheme renormalizes cubic lattice sections into qubits and connects them with CZ gates, assuming deterministic 3-photon GHZ states generated from Bell pairs.
Results
The proposal uses resources that scale linearly with computation size, supports computational depth 1500 for L = 6 before percolation probability falls below 90%, and requires at least an order of magnitude fewer resources than the compared scheme.
Takeaways & Limitations
Compared with the previous scheme, it uses at least 14% fewer Bell pairs and improves heralded loss tolerance by 5%.
Takeaways & Limitations
The scaling and renormalization analysis assumes GHZ and Bell pairs are available on demand, including deterministic 3-photon GHZ states.
Abstract
from arXiv · showhide
Single photons, manipulated using integrated linear optics, constitute a promising platform for universal quantum computation. A series of increasingly efficient proposals have shown linear-optical quantum computing to be formally scalable. However, existing schemes typically require extensive adaptive switching, which is experimentally challenging and noisy, thousands of photon sources per renormalized qubit, and/or large quantum memories for repeat-until-success strategies. Our work overcomes all these problems. We present a scheme to construct a cluster state universal for quantum computation, which uses no adaptive switching, no large memories, and which is at least an order of magnitude more resource-efficient than previous passive schemes. Unlike previous proposals, it is constructed entirely from loss-detecting gates and offers a robustness to photon loss. Even without the use of an active loss-tolerant encoding, our scheme naturally tolerates a total loss rate of $\sim 1.6\%$ in the photons detected in the gates. This scheme uses only 3-GHZ states as a resource, together with a passive linear-optical network. We fully describe and model the iterative process of cluster generation, including photon loss and gate failure. This demonstrates that building a linear optical quantum computer need be less challenging than previously thought.
CALCULATING THE PERCOLATION THRESHOLD FROM FINITE SIZE LATTICES
The paper estimates the percolation threshold for finite lattices using renormalisation and the lattice’s self-similarity at criticality. This avoids assuming a functional form for the finite-size percolation probability.
- Finite-size percolation probability: Π(p, L) is smooth for finite lattices, unlike the infinite-lattice step function at the percolation threshold.For an infinite lattice, Π is 0 below pc and 1 above pc; finite-size corrections smooth this transition.
- Renormalisation method: Renormalisation replaces lattice cells of linear size b with supersites when b is much smaller than the correlation length ξ.The correlation length is the typical cluster diameter and diverges at the percolation threshold.
- Renormalisation method: At criticality, lattice self-similarity gives Π(pc, L) = Π(pc, L/b), providing the condition used to identify the threshold.The renormalised lattice has the same properties as the original lattice at pc.
RENORMALISATION OF THE LATTICE AND SCALING OF RESOURCES
The scheme renormalises cubic lattice regions into qubits and connects them with CZ gates, producing a measurement-based computational structure. Its resource requirements scale linearly with computation size under the stated on-demand resource assumptions.
- Renormalisation procedure: CZ gates between qubits on cube sides implement gates in the renormalised lattice.The construction interprets the fused qubits as the logical structure used for measurement-based computation.
- Renormalisation procedure: Cubic pieces of the lattice are treated as renormalised qubits, with blue spanning clusters and disconnected regions determining available connections.Missing widthwise connections correspond to logical-qubit gates that can be delayed or reconfigured without posing a significant problem.
- Resource scaling: A renormalised qubit contains L^3 physical qubits, and a computation with n logical qubits and depth k uses n · k renormalised qubits.Here, k is the number of measurements per logical-qubit line in the MBQC model.
- Percolation performance: For L = 6, the probability remains above 90% through a computational depth of 1500, while larger L permits higher depth.The percolating-path probability decays exponentially with fused renormalised qubits, but the decay is negligible over O(100) qubits for L ≥6.
- Resource scaling: The total number of lattice sites is (n · k) L^3, requiring 3(n · k) L^3 3-photon GHZ states and 4(n · k) L^3 fusions.Each fusion uses 15 polarization rotators and 4 polarising beamsplitters at a 75% success rate.
- Resource scaling: With L treated as constant for large computations, resource dependence is linear in n · k.The paper identifies n and k as the variables that scale with computer size under on-demand 3-photon GHZ and Bell-pair assumptions.
COMPARISON WITH PREVIOUS PERCOLATION SCHEMES
The paper compares its ballistic construction with the previous percolation scheme of Kieling et al. It reports smaller renormalised qubits, lower Bell-pair consumption, and improved heralded-loss tolerance under matched comparison conditions.
- Comparison setup: The comparison uses data for Kieling et al.’s diamond-lattice block size k^3 versus renormalised square-lattice size L, selecting the (1.00, 0.5) dataset.The authors compare maximum reachable L values at corresponding k values with Π(L) ≥ 1/2.
- Renormalised-qubit size: The proposed scheme noticeably reduces the renormalised-qubit size relative to Kieling et al.’s scheme.The resource advantage becomes larger when Bell pairs per renormalised qubit and for the entire cluster are included.
- Resource accounting: 153 Bell pairs are consumed to generate a deterministic 4-photon GHZ state in the comparison scheme.The comparison assigns 3 Bell pairs per attempt for the 4-photon resource.
- Resource accounting: 42 Bell pairs are consumed to generate a deterministic 3-photon GHZ state under the paper’s comparison assumptions.The proposal assumes 2 Bell pairs per attempt and repeats generation to obtain a deterministic state.
- Bell-pair consumption: The proposed cluster construction uses at least an order of magnitude fewer resources than Kieling et al.’s scheme.The comparison counts Bell pairs needed per renormalised qubit and for an entire L × L renormalised cluster.
- Bell-pair consumption: 14% more Bell pairs are used by Kieling et al.’s scheme than by the proposed scheme for comparable cluster sizes.The ratio compares points with L values of the same order of magnitude.
- Loss robustness: The proposed construction improves heralded-loss tolerance by 5% relative to the previous approach.The paper presents this as an additional benefit beyond reduced resource consumption.